University of Toronto Mathematics NetworkAnswers and Explanations

Number Systems with Different Kinds of Numbers

In mathematics, the phrase number system refers
to any collection of objects
that has the following properties:

there is a rule for how to add two objects together,

there is a rule for how to multiply two objects together,

the rules for addition and multiplication satisfy the
familiar properties of
arithmetic, such as

commutativity (order doesn't matter),

associativity (in a sum of three of more terms, it
doesn't matter which two you add first, and likewise for products),
and

distributivity (a(b+c) = ab + ac), and

all objects have corresponding negatives (from which we
can define subtraction as the addition of one object to the negative of
another), and some objects have reciprocals (from which we can define
division as the multiplication of one object by the reciprocal of another).
The properties of subtraction and division follow automatically from
the corresponding properties of addition and multiplication, so we need not
mention them explicitly.

In some cases we
might consider two objects in the collection to be equal even if
they are not the same object; for instance, the fractions 1/2 and
2/4 are equal, even though they are different pairs of numbers. When
this is the case, we also need, in addition to the properties above,
a precise definition of when two objects are equal.

Roughly speaking, any collection of objects that satisfies these
properties is, by definition, a number system.

Note: this is only a rough statement. If you wanted to say
things more accurately, you'd have to take into account the fact that

strictly speaking, some of these properties need to be stated
a little more precisely than I stated them;

the phrase "number
system" is not often used; instead, one uses more fancy language such
as "field" or "commutative ring"; and

usually the rules for addition and multiplication satisfy
various other properties besides the ones I have listed, and one
uses terminology that precisely expresses which of these properties
are present.

The Natural Numbers. These are the
counting numbers 1, 2, 3, ... that are possible answers to the
question "how many?" They are abstract concepts that describe sizes
of sets.

The Integers. These are abstract concepts that describe,
not sizes of sets, but the relative sizes of two sets. They are
the possible answers to the question "how many more does A have than
B has?" They include both positive numbers (meaning A has more than B)
and negative numbers (meaning B has more than A).

The Rational Numbers. These are abstract concepts that
describe ratios of sizes of sets. They do not model sizes of
sets the way that natural numbers do. If you say "I ate 3/4 of a pie", you
are not saying that the set of things you ate had 3/4 elements. Instead, you
are expressing a ratio of two integer quantities: 3, the number
of pie-quarters that you
ate, and 4, the number of pie-quarters that make up a whole pie.

The Real Numbers. These are abstract concepts that
describe measurements of continuous quantities, such as length,
weight, quantity of fluid, etc. (Don't let the word "real"
fool you; the real numbers
are no more "real"
in the ordinary English sense of the word than are any other
kind of numbers.)

The Complex Numbers. These are pairs of real numbers, with
pairs of the form (x,0) behaving the same as ordinary real numbers x, but
with other pairs (whose second entry is non-zero) behaving differently.
The rule for multiplication is

(a, b) times (c, d) =
(ac-bd, ad+bc)

which means that the pair (0,1), when squared, gives (-1,0) which
in this context is considered to be
the same as the real number -1 (since a complex number of the form (x,0)
is indistinguishable by its arithmetic properties from the real number x,
we can consider it to be describing the same underlying concept, much
as we consider a fraction of the form x/1 to be describing the same
underlying concept as the integer x).

Thus, the complex numbers are a number system in which there is an object whose
square is -1. This object is denoted by i, and the pair
(a,b) is
written as "a + bi". When a=0, such a number
is called an "imaginary number".

Complex numbers are extremely important mathematically. They have
less direct relevance to real-world situations, since they aren't
measurements of single quantities. However, they relate directly to a few
real-world situations (such as the strength of an electromagnetic
field), and they relate indirectly to many more since many properties
of real numbers can be more easily understood in the larger context of
complex numbers.